Multi-scale structure integrated topological optimization design method

Abstract

The invention discloses a multi-scale structure integrated topological optimization design method. The method comprises the following steps: establishing a multi-scale finite element model; the microscopic density is initialized according to the set target volume fraction; the macroscopic density is calculated according to the volume average of the microcosmic density; the Young modulus obtained after material interpolation is substituted into the multi-scale finite element model, strain energy of each microcosmic unit is obtained, and the strain energy of the microcosmic units in the same macroscopic unit is summed to obtain strain energy of the macroscopic units; simplifying iterative updating of two scale design variables into iterative updating of one scale design variable by utilizing a coupling relationship between the macroscopic density and the microscopic density and a chain derivation rule; carrying out iterative updating on the microscopic density by adopting a gradient-based solver; and calculating the macroscopic density according to the microscopic density, and displaying the macroscopic scale and the optimization result of the microscopic scale. The method is simple in process, efficient in calculation and good in optimization performance, and provides technical support for design and application of a multi-scale structure.

Description

Technical Field

[0001] The present invention belongs to the technical field of multi-scale structure design, and specifically relates to a multi-scale structure integrated topology optimization design method. Background Art

[0002] Multiscale structures have the characteristics of light weight and high specific stiffness, which makes them have great application prospects in aerospace, automobile, medical and other fields. Multiscale structures are also considered to be an important development direction of the next generation of lightweight structures. With the advancement of additive manufacturing technology, people can manufacture complex multiscale structures. However, how to make use of the huge design space brought by multiscale structures and fully tap the design potential of multiscale structures has become a new problem. Topology optimization is an advanced structural design method that can maximize the performance of the structure under the given material volume constraint. Multiscale structural topology optimization design is to explore the application potential of materials from both macro and micro scales, and is an important way to develop high-performance, lightweight and multifunctional advanced structures. Driven by the huge demand for new high-performance ultra-light structures in engineering fields such as aerospace, and with the strong support of advanced manufacturing processes, topology optimization design for multiscale structures has gradually attracted more and more attention and research, and has become one of the hottest research directions in the field of structural topology optimization in recent years. However, the existing multiscale structural topology optimization design methods still have serious defects in terms of computational efficiency and optimized structural performance, which makes it impossible to achieve efficient and high-performance design of multiscale structures.

[0003] There are three main types of existing multi-scale structural topology optimization design methods: full-scale method, hierarchical multi-scale topology optimization method and concurrent multi-scale topology optimization method. The main defects of these three methods are as follows:

[0004] (1) The full-scale method is to establish a finite element model of the full-scale structure and use the same method as the single-scale topology optimization to optimize the structure. At the same time, periodic constraints or local volume constraints can also be used to promote the generation of multi-scale structures. Although this method is simple, it requires the establishment of a full-scale finite element model, which will make the finite element model too large and have too many design variables, resulting in high computational costs.

[0005] (2) The hierarchical multi-scale topology optimization method requires iterative solutions at both the macro and micro scales. At the same time, the number of local optimization problems at the micro scale is equal to the number of units at the macro scale. Therefore, this method still has the problem of high computational cost. Although local optimization problems can be solved by parallel computing, the efficiency of solving general problems is still low.

[0006] (3) The concurrent multiscale topology optimization method establishes two independent sets of design variables at the macro and micro scales, and updates the two sets of design variables simultaneously. Although this method can optimize the macro and micro structures simultaneously, it is essentially still solving the two scales independently, so the volume fraction of each representative volume element at the micro scale needs to be manually given. Since it is impossible to predict the optimal volume fraction of each representative volume element, it is often difficult for multiscale structures designed based on this method to achieve optimal performance.

[0007] Therefore, given the wide application prospects of multi-scale structures and the urgent need for multi-scale structure design, it is urgent to establish a simple, efficient and high-performance multi-scale structure topology optimization design method. Summary of the invention

[0008] In response to the above-mentioned technical problems, the present invention discloses a multi-scale structure integrated topology optimization design method with simple process, efficient calculation and good optimization performance, which provides technical support for the design and application of multi-scale structures and can meet the use requirements of lightweight and high-performance structures in aerospace, automotive, medical and other fields.

[0009] Technical solution:

[0010] A multi-scale structure integrated topology optimization design method, the multi-scale structure integrated topology optimization design method comprising the following steps:

[0011] S1, based on Direct FE 2 The model discretizes the design domain and establishes a multi-scale finite element model; the multi-scale finite element model includes a finite element mesh of the macro model obtained by finite element discretization of the macro model of the design domain, and a finite element mesh of the micro model obtained by finite element discretization of the micro model including the initial design established at the integration point of each macro unit; the unit in the macro mesh is called a macro unit, the unit in the micro mesh is called a micro unit, and the micro model is defined as a representative volume element;

[0012] S2, taking the density of the micro-unit as the micro-design variable and initializing the micro-density according to the set target volume fraction;

[0013] S3, taking the density of the macro unit as the macro design variable, the macro density is calculated based on the volume average of the micro density;

[0014] S4, interpolating the macro density and micro density respectively, multiplying the interpolated macro density, the interpolated micro density and the Young's modulus of the material to obtain the interpolated Young's modulus of the material; substituting the interpolated Young's modulus of the material into the multi-scale finite element model for multi-scale finite element analysis to obtain the strain energy of each micro unit; summing the strain energies of the micro units in the same macro unit to obtain the strain energy of the macro unit;

[0015] S5, constructing a structural objective function model and a structural volume model, and calculating the micro-sensitivity and macro-sensitivity of the objective function and the volume respectively; using a macro-filter radius and a micro-filter radius to filter the macro-sensitivity and micro-sensitivity of the objective function respectively, using the strain energy of the macro-unit and the strain energy of the micro-unit to represent the filtering results, and converting the filtering of the macro-sensitivity and micro-sensitivity of the objective function into filtering of the strain energy of the macro-unit and the strain energy of the micro-unit; summing the filtered objective function sensitivities of the same micro-unit position in the same macro-unit representative volume element, and summing the volume sensitivities of the same micro-unit position in the same macro-unit representative volume element, to obtain the summed objective function sensitivity and volume sensitivity;

[0016] S6, inputting the summed objective function sensitivity and volume sensitivity into the gradient-based optimization solver, iteratively updating the microscopic density, and then repeating the steps of S3-S6 until the error is less than the preset error threshold, stopping the iteration, and obtaining the optimized microscopic density;

[0017] S7, calculating the volume average of the optimized microscopic density to obtain the optimized macroscopic density, and displaying the optimized macroscopic density and microscopic density simultaneously through a density cloud map to obtain the optimization result.

[0018] Step S1 further comprises:

[0019] S11, discretize the macro model of the design domain by finite element to obtain the finite element mesh of the macro model, the unit in the finite element mesh of the macro model is called a macro unit, and the node in the mesh is called a macro node;

[0020] S12, establishing a microscopic model including the initial design at the integration point of each macroscopic unit and performing finite element discretization to obtain a finite element mesh of the microscopic model, wherein the unit in the finite element mesh of the microscopic model is called a microscopic unit, and the node of the mesh is called a microscopic node;

[0021] S13, establish the displacement constraint relationship between the macro model and the micro model:

[0022]

[0023] in is the displacement of the microscopic node, the subscripts R, L, T and B represent the nodes on the right, left, top and bottom sides of the volume element, respectively; x and y are the coordinates in the rectangular coordinate system, and u is the displacement of the macroscopic unit; and are the lengths of the volume element in the x and y directions respectively;

[0024] S14, the microscopic model is enlarged and a negligible Young's modulus is assigned to the material of the macroscopic model so that the overall energy of the structure is conserved; the enlargement factor of the microscopic model is w α J α / V α , w α and J α denote the weight and Jacobian of Gaussian integration point α, V α is the volume of the representative volume element at the Gaussian integration point α;

[0025] S15, establishing the boundary conditions of the entire model according to the actual working conditions of the structure, including displacement boundary conditions, external force boundary conditions, etc.;

[0026] S16, calculate the volume of each microscopic unit

[0027] Furthermore, in step S2, the initial density values ​​of all microscopic units are set to v f It is the target volume fraction that is set; when the density value is 1, it means that the unit is retained, and when the density value is 0, it means that the unit is deleted.

[0028] In step S3, the macroscopic density is calculated according to the following formula:

[0029]

[0030] where ρ e is the density of macro unit e, V e is the volume of the macro unit e, defined as Microscopic unit The density of Microscopic unit Volume; It represents the sum of all microscopic units within the macroscopic unit e.

[0031] Step S4 further comprises:

[0032] S41, using solid isotropic material penalty method to penalize microscopic density Interpolate to obtain the microscopic density after interpolation Where p is the penalty factor for the macroscopic density ρ e Perform linear interpolation without penalty;

[0033] S42, multiply the interpolated macroscopic density, the interpolated microscopic density and the Young's modulus of the material to obtain the interpolated Young's modulus of the material:

[0034]

[0035] in is the interpolated Young's modulus, is the Young's modulus given for the material;

[0036] S43, the Young's modulus after material interpolation is substituted into the multi-scale finite element model, and a multi-scale finite element analysis is performed to obtain the strain energy of each micro-unit:

[0037]

[0038] in Microscopic unit The strain energy, For unit The node displacement vector, According to Direct FE 2 The unit stiffness matrix after amplification by the method;

[0039] S44, sum the strain energies of the micro-units in the same macro-unit to obtain the strain energy of the macro-unit:

[0040]

[0041] Among them U e is the strain energy of the macro unit e, α is the number of the integration point in the macro unit, (*) e Indicates the variables in the macro unit e, (*) α represents the variable within the integration point α.

[0042] Furthermore, the multi-scale structure integrated topology optimization design method according to claim 1 is characterized in that, in step S5, the process of respectively calculating the microscopic sensitivity and macroscopic sensitivity of the objective function and the volume comprises the following steps:

[0043] A51, calculate the partial derivatives of the objective function and volume with respect to the microscopic design variables, and obtain the microscopic sensitivity of the objective function and volume;

[0044] A52, according to the chain rule, calculate the partial derivatives of the objective function and volume with respect to the macro design variable, obtain the macro sensitivity of the objective function and volume, and multiply the macro sensitivity of the objective function and volume by the partial derivatives of the macro design variable with respect to the micro design variable;

[0045] A53, adding the calculation results of step A51 and step A52 to obtain the sensitivity of the target function and volume.

[0046] Step S5 further comprises the following steps:

[0047] Construct a structural objective function model:

[0048]

[0049] Where c is the objective function of the structure, is the vector of all microscopic densities, is the overall displacement vector of the microscopic node, According to Direct FE 2 The overall stiffness matrix of the structure after amplification by the method;

[0050] The sensitivity of the objective function is calculated according to the chain derivation rule:

[0051]

[0052] in,

[0053] According to the adjoint method, we get:

[0054]

[0055] The sensitivity of the objective function is obtained as:

[0056]

[0057] Build the structural volume model:

[0058]

[0059] in The microscopic density is The total volume of the structure;

[0060] The sensitivity of volume is calculated using the chain rule:

[0061]

[0062] Taking the derivative and simplifying we get:

[0063]

[0064] Furthermore, the filtering process of the macro unit strain energy and the micro unit strain energy includes the following steps:

[0065] The macro unit strain energy is filtered according to the macro filtering radius, and the ratio of the macro unit strain energy before and after filtering is multiplied by the micro unit strain energy to obtain the scaled micro unit strain energy; the representative volume element is periodically expanded, and the scaled micro unit strain energy is filtered according to the relative position relationship between the micro units in the expanded representative volume element and the micro filtering radius.

[0066] Furthermore, the macro filter radius and micro filter radius are used to respectively calculate the macro sensitivity of the objective function. and microscopic sensitivity To filter:

[0067]

[0068] where w ka and is the filter weight:

[0069] w ka =max(0,r mac -r ka );

[0070]

[0071] In the formula, r mac and r mic are the macro filter radius and micro filter radius respectively, r ka is the distance between the center points of macro unit k and macro unit a, Microscopic unit Micro-unit The distance between the center points; the filtering formula expressed by strain energy is:

[0072]

[0073] in and Represents macro unit k and micro unit respectively Filtered strain energy:

[0074]

[0075] For irregular elements, the modified filtered result for the strain energy density is:

[0076]

[0077] Where V k 、V a is the volume of the macro unit, is the volume of the microscopic unit.

[0078] After filtering the macro unit strain energy, the micro unit strain energy is scaled according to the change of the macro unit strain energy. The scaling factor is When filtering the microscopic strain energy, the representative volume element is periodically expanded, and the scaled microscopic unit strain energy is filtered according to the relative position relationship between the microscopic units in the expanded representative volume element and the microscopic filtering radius;

[0079] The filtered objective function sensitivities of the same micro-unit position in the representative volume element in the same macro-unit are summed, and the volume sensitivities of the same micro-unit position in the representative volume element in the same macro-unit are summed; the summed objective function sensitivity and volume sensitivity are obtained:

[0080]

[0081] The symbol <*> α It means summing the integration points α within the macro unit e.

[0082] Furthermore, in step S6, the error err is calculated at each iteration step update:

[0083]

[0084] Where k′ is the current iteration step, c k′-i+1 is the flexibility at the k′-i+1th step, and N is the length of the average interval.

[0085] Beneficial effects:

[0086] First, the multi-scale structure integrated topology optimization design method of the present invention transforms the multi-scale optimization problem into a single-scale optimization problem for solution by utilizing the connection between design variables of different scales, thereby simplifying the design process to a level comparable to that of the full-scale method.

[0087] Second, the multi-scale structure integrated topology optimization design method of the present invention is similar to the hierarchical multi-scale topology optimization. This method only requires an overall volume fraction constraint and does not require a local volume constraint. This promotes the adaptive distribution of materials between different representative volume elements and different scales, and gives full play to the design potential of multi-scale structures.

[0088] Third, the multi-scale structure integrated topology optimization design method of the present invention only requires one loop to solve, without nested loops, and the optimization solver only needs to update the microscopic density, thereby reducing the computational cost of updating the design variables to a level comparable to concurrent multi-scale topology optimization. BRIEF DESCRIPTION OF THE DRAWINGS

[0089] Figure 1 It is a flow chart of the multi-scale structure integrated topology optimization design method of the present invention;

[0090] Figure 2 A multi-scale integrated topology optimization design framework diagram provided by the present invention;

[0091] Figure 3 Schematic diagram of periodic expansion of representative volume elements when filtering microscopic strain energy;

[0092] Figure 4 It is a schematic diagram of the bridge structure;

[0093] Figure 5 Direct FE for bridge structure 2 Model schematic diagram;

[0094] Figure 6 It is a schematic diagram of the flexibility change curve during the bridge structure optimization process;

[0095] Figure 7 Schematic diagram of bridge structure optimization results (r mac =0mm, r mic =0.2mm);

[0096] Figure 8 Schematic diagram of bridge structure optimization results (r mac =0mm, r mic =0.3mm);

[0097] Fig. 9 Schematic diagram of bridge structure optimization results (r mac =30mm, r mic =0.2mm);

[0098] Fig.10 Schematic diagram of bridge structure optimization results (r mac =30mm, r mic =0.3mm);

[0099] Fig.11 Schematic diagram of topology optimization design for multi-condition and multi-scale structures;

[0100] Fig.12 Direct FE for multi-scale structures 2 Model schematic diagram;

[0101] Fig.13 Schematic diagram of multi-scale structural optimization results under multiple working conditions (r mac =3mm, r mic =0.2mm);

[0102] Fig.14 Schematic diagram of multi-scale structural optimization results under multiple working conditions (r mac =30mm, r mic =0.2mm). DETAILED DESCRIPTION

[0103] The following examples will enable those skilled in the art to more fully understand the present invention, but are not intended to limit the present invention in any way.

[0104] The present invention discloses a multi-scale structure integrated topology optimization design method, which comprises the following steps:

[0105] S1, based on Direct FE 2 The model discretizes the design domain and establishes a multi-scale finite element model; the multi-scale finite element model includes a finite element mesh of a macro model obtained by finite element discretization of a macro model of the design domain, and a finite element mesh of a micro model obtained by finite element discretization of a micro model including an initial design established at an integral point of each macro unit;

[0106] S2, taking the density of the micro-unit as the micro-design variable and initializing the micro-density according to the set target volume fraction;

[0107] S3, taking the density of the macro unit as the macro design variable, the macro density is calculated based on the volume average of the micro density;

[0108] S4, interpolating the macro density and micro density respectively, multiplying the interpolated macro density, the interpolated micro density and the Young's modulus of the material to obtain the interpolated Young's modulus of the material; substituting the interpolated Young's modulus of the material into the multi-scale finite element model for multi-scale finite element analysis to obtain the strain energy of each micro unit; summing the strain energies of the micro units in the same macro unit to obtain the strain energy of the macro unit;

[0109] S5, construct the structural objective function model and the structural volume model. Taking the maximization of structural stiffness as an example, the objective function is the flexibility of the structure. The sensitivity of the objective function and the sensitivity of the volume are calculated. Since the macro design variable is a function of the micro design variable, only the sensitivity to the micro design variable needs to be calculated when calculating the sensitivity, thereby simplifying the iterative update of the two scale design variables to the iterative update of the one scale design variable. And the sensitivity is the addition of two parts: the first part is to first calculate the partial derivative of the macro density according to the chain derivation rule, and then multiply it with the derivative of the macro density to the micro density, and the second part is the partial derivative of the micro density; different filter radii are used to filter the macro sensitivity and micro sensitivity respectively, where the macro sensitivity is the partial derivative of the objective function to the macro density, and the micro sensitivity is the partial derivative of the objective function to the micro density. The strain energy of the micro unit and the strain energy of the macro unit can be used to represent the filtering result. After filtering the strain energy of the macro unit, the strain energy of the micro unit is scaled according to the ratio of the strain energy of the macro unit before and after filtering. At the same time, when filtering the microscopic strain energy, the representative volume element needs to be expanded periodically; since the representative volume elements in the macro unit are distributed periodically, that is, the density values ​​of the micro units at the same position in each representative volume element in the same macro unit are the same, the sensitivities of the micro units at the same position in the representative volume element in the macro unit are summed;

[0110] S6, inputting the summed objective function sensitivity and volume sensitivity into the gradient-based optimization solver, iteratively updating the microscopic density, and then repeating the steps of S3-S6 until the error is less than the preset error threshold, stopping the iteration, and obtaining the optimized microscopic density;

[0111] S7, calculates the macro density based on the micro density obtained in the last iteration step, and displays the optimization results of the macro and micro scales simultaneously through the density cloud map.

[0112] See also Figure 1 and Figure 2 The specific implementation process of this method is as follows:

[0113] 1. Establishing a multi-scale finite element model

[0114] Based on Direct FE 2 The design domain is discretized by this method, and a multi-scale finite element model is established, which specifically includes the following contents:

[0115] (1) The macro model of the design domain is discretized by finite element method to obtain a finite element mesh of the macro model, in which the unit is called a macro unit and the node is called a macro node.

[0116] (2) A microscopic model including the initial design is established at the integration point of each macroscopic unit and finite element discretization is performed to obtain a finite element mesh of the microstructure, in which the units are called microscopic units and the nodes are called microscopic nodes.

[0117] (3) Establish the displacement constraint relationship between the macro model and the micro model, which typically includes periodic boundary conditions. Taking the two-dimensional problem as an example, the formula for the periodic boundary condition is as follows:

[0118]

[0119] in is the displacement of the microscopic node, the subscripts R, L, T and B represent the nodes on the right, left, top and bottom of the volume element, respectively. x and y are the coordinates in the rectangular coordinate system, and u is the displacement of the macroscopic unit. and are the lengths of the volume element in the x and y directions respectively.

[0120] (4) The microscopic model is enlarged and a negligible Young's modulus is assigned to the material of the macroscopic model so that the overall energy of the structure is conserved. The enlargement factor of the microscopic model is w α J α / V α , where w α and J αdenote the weight and Jacobian of Gaussian integration point α, V α is the volume of the volume element at the Gaussian integration point α. The negligible Young's modulus is a very small number, for example: 1×10 -9 E0, where E0 is the Young's modulus of the material.

[0121] (5) Establish the overall boundary conditions of the model according to the actual working conditions of the structure, including displacement boundary conditions, external force boundary conditions, etc.

[0122] (6) Store the volume of each microscopic unit

[0123] 2. Initialize Micro Design Variables

[0124] In the density-based topology optimization method, the design variable refers to the pseudo "density" artificially defined for each unit. When the density value is 1, it means that the unit is retained, and when the density value is 0, it means that the unit is deleted. The micro-design variable refers to the density of the micro-unit, which can also be called micro-density. According to the set target volume fraction v f Initialize the value of micro density. The initial density of all micro units can be set to the same value. At the same time, in order to ensure that the initial volume fraction is the same as the target volume fraction, the initial density of all micro units is set to

[0125] 3. Calculation of Macro Design Variables

[0126] The macro design variable refers to the density of the macro unit, which can also be called macro density. The macro density is equal to the volume average of the micro density, so the macro density can be calculated according to the following formula:

[0127]

[0128] where ρ e is the density of macro unit e, V e is the volume of the macro unit e, defined as Microscopic unit The density of Microscopic unit volume. It represents the sum of all microscopic units within the macroscopic unit e.

[0129] 4. Multi-scale finite element analysis

[0130] First, the macroscopic density ρ e and microscopic density The micro density can be interpolated using the classic SIMP method (i.e., solid isotropic material penalty method). The micro density interpolation is Where p is the penalty factor, which is generally taken as 3. For the macroscopic density, only linear interpolation is required without penalty. This is because the penalty for density is to suppress the appearance of intermediate density units, so that the optimization result tends to a discrete 0-1 design. However, the intermediate density of the macroscopic unit represents that the microscopic density of the internal volume element is not all 0 or 1, which means that there is a microstructure in the RVE. Therefore, there is no need to penalize the intermediate density of the macroscopic unit, and the Young's modulus and volume of the structure are linearly related to the macroscopic density. Multiply the interpolated macroscopic density, the interpolated microscopic density, and the Young's modulus of the material to obtain the interpolated Young's modulus of the material:

[0131]

[0132] in is the interpolated Young's modulus, is the Young's modulus given by the material. Substitute the Young's modulus after material interpolation into the Direct FE established in step 1 2 Model, perform multi-scale finite element analysis, and obtain the strain energy of each micro unit:

[0133]

[0134] in Microscopic unit The strain energy, For unit The node displacement vector, According to Direct FE 2 The unit stiffness matrix after amplification by the method.

[0135] Then the macroscopic unit strain energy is calculated based on the microscopic unit strain energy:

[0136]

[0137] Among them U e is the strain energy of the macro unit e, α is the number of the integration point in the macro unit, (*) e Indicates the variables in the macro unit e, (*) α represents the variable within the integration point α.

[0138] V. Sensitivity Analysis

[0139] For the design problem of minimizing flexibility (i.e. maximizing stiffness), the flexibility of the structure can be written as follows:

[0140]

[0141] Where c is the flexibility of the structure, is the vector of all microscopic densities, is the overall displacement vector of the microscopic node, According to Direct FE 2 The overall stiffness matrix of the structure after the method is enlarged. According to the chain derivation rule, the sensitivity of flexibility can be calculated as:

[0142]

[0143] in

[0144]

[0145] At the same time, according to the adjoint method, we can get:

[0146]

[0147] Finally, the sensitivity of flexibility can be obtained:

[0148]

[0149] The volume of the structure can be expressed as:

[0150]

[0151] in The microscopic density is The total volume of the structure when . The sensitivity of the volume can also be calculated using the chain rule:

[0152]

[0153] After simple derivation and simplification, we can get:

[0154]

[0155] In order to solve the mesh dependency, the flexibility sensitivity is filtered. Considering the difference in length scale between macrostructure and microstructure, different filter radii are needed to filter the macro sensitivity. and microscopic sensitivity Filter separately.

[0156]

[0157] where w ka and is the filter weight, defined as follows:

[0158] w ka =max(0,r mac -r ka )

[0159]

[0160] where r mac and r mic are the macro filter radius and micro filter radius respectively, r ka is the distance between the center points of macro unit k and macro unit a, Microscopic unit Micro-unit The above filtering formula can be expressed by strain energy as:

[0161]

[0162] in and Represents macro unit k and micro unit respectively Filtered strain energy:

[0163]

[0164] For irregular units, the above formula should be modified to filter the strain energy density:

[0165]

[0166] Where V k 、V a is the volume of the macro unit, is the volume of the microscopic unit.

[0167] It should be noted that due to the Direct FE 2 In the model, the macro unit itself does not have strain energy. Here, the strain energy of the macro unit is obtained by adding the strain energy of the micro unit. Therefore, after filtering the strain energy of the macro unit, the strain energy of the micro unit needs to be adjusted accordingly. In practical applications, a feasible strategy is to scale the strain energy of the micro unit according to the change of the strain energy of the macro unit. The scaling factor is At the same time, since the representative volume elements are periodically distributed in the macro unit, when filtering the micro strain energy, the representative volume elements need to be periodically expanded, such as Figure 3 shown.

[0168] Also, since the representative volume elements are distributed periodically in the macro unit, that is, the density value of the micro units at the same position in each representative volume element in the same macro unit is the same. Therefore, it is necessary to sum the sensitivity of the micro units at the same position in the representative volume element in the macro unit. Finally, the sensitivity of the multi-scale integrated topology optimization method is:

[0169]

[0170] The symbol <*> α It means summing the integration points α within the macro unit e.

[0171] 6. Design variable update

[0172] A gradient-based solver (e.g., moving asymptote method) is used to update the design variables. Since the macroscopic density is a function of the microscopic density, the macroscopic density is a dependent variable, while the microscopic density is an independent variable. Therefore, the solver only needs to update the microscopic density and then calculate the macroscopic density according to the formula. The error err is calculated at each iteration update:

[0173]

[0174] Where k is the current iteration step, c k′-i+1 is the flexibility at the k′-i+1th step, N is the length of the average interval, which is set to 5 here, and the iteration stops when err<0.001.

[0175] 7. Output optimization results

[0176] The macroscopic density is calculated based on the microscopic density obtained in the last iteration step, and the optimization results of the macroscopic and microscopic scales are simultaneously displayed through the density cloud map.

[0177] It should be noted that the present invention utilizes the coupling relationship between macro-design variables and micro-design variables and the chain differentiation rule to simplify the iterative update of two-scale design variables into the iterative update of one-scale design variable. The method is not limited to the optimization problem of minimizing flexibility, but can solve the optimization problems of most multi-scale structures.

[0178] Example 1: Multi-scale topology optimization design of bridge structure

[0179] Bridge structure such as Figure 4 As shown, the midpoint of its bottom edge bears a downward concentrated force F = 2 N. The geometric dimensions of the structure are L = 800 mm, H = 400 mm.

[0180] Due to the structural symmetry, only half of the structure is modeled, e.g. Figure 5 As shown in Figure 1. The model imposes symmetric boundary conditions on the left boundary, constrains the vertical displacement at the lower right corner, and applies a downward concentrated force of F / 2 at the lower left corner. The model is discretized into 20×20 macro units. At the four integration points of each macro unit, a representative volume element model is established and discretized into 160 units. The size of RVE is The initial design was a circular hole with a diameter of d=0.8 mm.

[0181] The optimization objective function is to maximize the structural stiffness, that is, to minimize the structural flexibility, and the volume fraction is v f =0.2. Four different combinations of macro and micro filter radii are used, namely: (1) r mac =0mm, r mic =0.2mm; (2)r mac =0mm, r mic =0.3mm; (3)r mac =30mm, r mic = 0.2 mm; and (4) r mac =30mm, r mic = 0.3 mm. The optimization results are then compared with those using the full-scale method. Figure 6 The multi-scale integration method of the present invention is compared with the full-scale method to compare the change in structural flexibility during the optimization process. As shown in the figure, the flexibility of the multi-scale integration method steadily decreases with the increase in the number of iterations, and finally converges after about 60-70 iterations. It can be seen from the figure that for different filter radii, the flexibility of the multi-scale integration method after convergence is always lower than that of the full-scale method, which proves the excellent design performance of the multi-scale integration method. Specifically, compared with the full-scale method, the multi-scale integration method achieves 3.6% (r mac =30mm, r mic =0.3mm) to 15.3% (r mac =0mm, r mic =0.3mm). This is because the multi-scale integration method can achieve adaptive distribution of materials between different scales and different representative volume elements.

[0182] Figures 7 to 10 The optimization results of the multi-scale integration method under different filter radii are shown, where each figure is divided into two parts. The left half is the macroscopic density ρ e , the right half is the microscopic density Since the macroscopic density is equal to the volume average of the microscopic density, the left half shows a range of intermediate density values. Meanwhile, the right half shows the detailed microstructure. In addition, five representative local areas in the right half are enlarged to show the optimization results more clearly. Figures 7 to 10It can be seen from the figure that: (1) the multi-scale integration method can optimize the macrostructure and microstructure at the same time, and the optimized structure is reasonably distributed along the principal stress direction at both scales; (2) for different representative volume elements, not only the microstructure is different, but also the volume fraction is different. Therefore, the multi-scale integration method can realize the adaptive distribution of materials between different representative volume elements; (3) changing the filter radius of one scale will have a great impact on the optimization result of another scale, so the multi-scale integration method can realize the adaptive distribution of materials between different scales; (4) in the absence of macro-scale filtering (r mac =0 mm), the grid dependence leads to a porous trend on the macro scale, which is consistent with the structural optimization theory, that is, the theoretically optimal stiffness structure exhibits multi-scale characteristics. This reveals the excellent multi-scale structural optimization potential of the multi-scale integration method. (5) The multi-scale integration method can adjust the distribution of materials between different scales by simply adjusting the filter radius of different scales.

[0183] Example 2: Multi-condition and multi-scale topology optimization design

[0184] Similar to other topology optimization methods, the multiscale integration approach can be easily extended to handle multiple load cases. Fig.11 As shown, the left side of the structure is fixed, and there are two load conditions: the top of the right side bears the upward unit concentrated force F1 and the bottom of the right side bears the downward unit concentrated force F2. The structural size is L = 400mm. Fig.12 Direct FE shown 2 Finite element model, the left side of the model is fixed, and the upper right corner and lower right corner points bear upward and downward unit concentrated forces respectively. The optimization objective function is to maximize the structural stiffness, that is, to minimize the structural flexibility, and the volume fraction is 0.2. Microscopic filter radius r mic is 0.2mm, the macro filter radius r mac Take them as 0mm and 30mm respectively.

[0185] Fig.13 and Fig.14 The optimization results under two filter radii are shown. Fig.13 and Fig.14 It can be seen that due to the symmetry of the two load conditions, the optimized structure also shows symmetry. At the same time, both the macrostructure and the microstructure are distributed along the principal stress direction. It can also be seen that when the macro filter radius increases, the branches of the macro structure become thicker and the small pores in the macro structure disappear. This shows that the multi-scale integration method can effectively control the optimization results of structures of different scales through the filter radius. At the same time, in the place where the concentrated load acts ( Fig.13 and Fig.14The material is completely preserved (the red box area in the figure), avoiding large deformation caused by stress concentration, demonstrating the ability of this method to adaptively distribute materials.

[0186] The above are only preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, some improvements and modifications without departing from the principle of the present invention should be regarded as the protection scope of the present invention.

Claims

1. A multi-scale structure integrated topology optimization design method, characterized in that: The multi-scale structure integrated topology optimization design method comprises the following steps: S1, based on Direct FE 2 The model discretizes the design domain and establishes a multi-scale finite element model; the multi-scale finite element model includes a finite element mesh of the macro model obtained by finite element discretization of the macro model of the design domain, and a finite element mesh of the micro model obtained by finite element discretization of the micro model including the initial design established at the integration point of each macro unit; the unit in the macro mesh is called a macro unit, the unit in the micro mesh is called a micro unit, and the micro model is defined as a representative volume element; S2, taking the density of the micro-unit as the micro-design variable and initializing the micro-density according to the set target volume fraction; S3, taking the density of the macro unit as the macro design variable, the macro density is calculated based on the volume average of the micro density; S4, interpolating the macro density and micro density respectively, multiplying the interpolated macro density, the interpolated micro density and the Young's modulus of the material to obtain the interpolated Young's modulus of the material; substituting the interpolated Young's modulus of the material into the multi-scale finite element model for multi-scale finite element analysis to obtain the strain energy of each micro unit; summing the strain energies of the micro units in the same macro unit to obtain the strain energy of the macro unit; S5, constructing a structural objective function model and a structural volume model, and calculating the micro-sensitivity and macro-sensitivity of the objective function and the volume respectively; using a macro-filter radius and a micro-filter radius to filter the macro-sensitivity and micro-sensitivity of the objective function respectively, using the strain energy of the macro-unit and the strain energy of the micro-unit to represent the filtering results, and converting the filtering of the macro-sensitivity and micro-sensitivity of the objective function into filtering of the strain energy of the macro-unit and the strain energy of the micro-unit; summing the filtered objective function sensitivities of the same micro-unit position in the same macro-unit representative volume element, and summing the volume sensitivities of the same micro-unit position in the same macro-unit representative volume element, to obtain the summed objective function sensitivity and volume sensitivity; S6, inputting the summed objective function sensitivity and volume sensitivity into the gradient-based optimization solver, iteratively updating the microscopic density, and then repeating the steps of S3-S6 until the error is less than the preset error threshold, stopping the iteration, and obtaining the optimized microscopic density; S7, calculating the volume average of the optimized microscopic density to obtain the optimized macroscopic density, and displaying the optimized macroscopic density and microscopic density simultaneously through a density cloud map to obtain the optimization result.

2. The multi-scale structure integrated topology optimization design method according to claim 1, characterized in that: Step S1 further comprises: S11, discretize the macro model of the design domain by finite element to obtain the finite element mesh of the macro model, the unit in the finite element mesh of the macro model is called a macro unit, and the node in the mesh is called a macro node; S12, establishing a microscopic model including the initial design at the integration point of each macroscopic unit and performing finite element discretization to obtain a finite element mesh of the microscopic model, wherein the unit in the finite element mesh of the microscopic model is called a microscopic unit, and the node of the mesh is called a microscopic node; S13, establish the displacement constraint relationship between the macro model and the micro model: in is the displacement of the microscopic node, the subscripts R, L, T and B represent the nodes on the right, left, top and bottom sides of the volume element, respectively; x and y are the coordinates in the rectangular coordinate system, and u is the displacement of the macroscopic unit; and are the lengths of the volume element in the x and y directions respectively; S14, the microscopic model is enlarged and a negligible Young's modulus is assigned to the material of the macroscopic model so that the overall energy of the structure is conserved; the enlargement factor of the microscopic model is w α J α / V α , w α and J α denote the weight and Jacobian of Gaussian integration point α, V α is the volume of the representative volume element at the Gaussian integration point α; S15, establishing the boundary conditions of the entire model according to the actual working conditions of the structure, including displacement boundary conditions, external force boundary conditions, etc.; S16, calculate the volume of each microscopic unit 3. The multi-scale structure integrated topology optimization design method according to claim 1, characterized in that: In step S2, the initial density values ​​of all microscopic units are set to v f is the set target volume fraction; When the density value is 1, it means that the cell is retained, and when the density value is 0, it means that the cell is deleted.

4. The multi-scale structure integrated topology optimization design method according to claim 1, characterized in that: In step S3, the macroscopic density is calculated according to the following formula: where ρ e is the density of macro unit e, V e is the volume of the macro unit e, defined as Microscopic unit The density of Microscopic unit Volume; It represents the sum of all microscopic units within the macroscopic unit e.

5. The multi-scale structure integrated topology optimization design method according to claim 1, characterized in that: Step S4 further comprises: S41, using solid isotropic material penalty method to penalize microscopic density Interpolate to obtain the interpolated microscopic density Where p is the penalty factor, for the macroscopic density ρ e Perform linear interpolation without penalty; S42, multiply the interpolated macroscopic density, the interpolated microscopic density and the Young's modulus of the material to obtain the interpolated Young's modulus of the material: in is the interpolated Young's modulus, is the Young's modulus given for the material; S43, the Young's modulus after material interpolation is substituted into the multi-scale finite element model, and a multi-scale finite element analysis is performed to obtain the strain energy of each micro-unit: in Microscopic unit The strain energy, For unit The node displacement vector, According to Direct FE 2 The unit stiffness matrix after amplification by the method; S44, sum the strain energies of the micro-units in the same macro-unit to obtain the strain energy of the macro-unit: Among them U e is the strain energy of the macro unit e, α is the number of the integration point in the macro unit, (*) e Indicates the variable in the macro unit e, (*) α represents the variable within the integration point α.

6. The multi-scale structure integrated topology optimization design method according to claim 1, characterized in that: The multi-scale structural integrated topology optimization design method according to claim 1 is characterized in that in step S5, the process of respectively calculating the micro-sensitivity and macro-sensitivity of the objective function and the volume comprises the following steps: A51, calculate the partial derivatives of the objective function and volume with respect to the microscopic design variables, and obtain the microscopic sensitivity of the objective function and volume; A52, respectively calculate the partial derivatives of the objective function and volume with respect to the macroscopic design variables, obtain the macroscopic sensitivities of the objective function and volume, and multiply the macroscopic sensitivities of the objective function and volume by the partial derivatives of the macroscopic design variables with respect to the microscopic design variables according to the chain rule; A53, adding the calculation results of step A51 and step A52 to obtain the sensitivity of the target function and volume.

7. The multi-scale structure integrated topology optimization design method according to claim 1, characterized in that: Step S5 further comprises the following steps: Construct a structural objective function model: Where c is the objective function of the structure, is the vector of all microscopic densities, is the overall displacement vector of the microscopic node, According to Direct FE 2 The overall stiffness matrix of the structure after amplification by the method; The sensitivity of the objective function is calculated according to the chain derivation rule: in, According to the adjoint method, we get: The sensitivity of the objective function is obtained as: Build a volumetric model of the structure: in The microscopic density is The total volume of the structure; The sensitivity of volume is calculated using the chain rule: Taking the derivative and simplifying we get:

8. The multi-scale structure integrated topology optimization design method according to claim 1, characterized in that: The filtering process of macro-element strain energy and micro-element strain energy includes the following steps: The macro unit strain energy is filtered according to the macro filtering radius, and the ratio of the macro unit strain energy before and after filtering is multiplied by the micro unit strain energy inside the macro unit to obtain the scaled micro unit strain energy; the representative volume element is periodically expanded, and the scaled micro unit strain energy is filtered according to the relative position relationship between the micro units in the expanded representative volume element and the micro filtering radius.

9. The multi-scale structure integrated topology optimization design method according to claim 1, characterized in that: The macroscopic sensitivity of the objective function is calculated by using the macroscopic filter radius and the microscopic filter radius. and microscopic sensitivity To filter: where w ka and is the filter weight: w ka =max(0,r mac -r ka ); In the formula, r mac and r mic are the macro filter radius and micro filter radius respectively, r ka is the distance between the center points of macro unit k and macro unit a, Microscopic unit Micro-unit The distance between the center points; the filtering formula expressed by strain energy is: in and Represents macro unit k and micro unit respectively Filtered strain energy: For irregular elements, the modified filtered result for the strain energy density is: Where V k 、V a is the volume of the macro unit, is the volume of the microscopic unit. After filtering the macro unit strain energy, the micro unit strain energy is scaled according to the change of the macro unit strain energy. The scaling factor is When filtering the microscopic strain energy, the representative volume element is periodically expanded, and the scaled microscopic unit strain energy is filtered according to the relative position relationship between the microscopic units in the expanded representative volume element and the microscopic filtering radius; The filtered objective function sensitivities of the same micro-unit position in the representative volume element in the same macro-unit are summed, and the volume sensitivities of the same micro-unit position in the representative volume element in the same macro-unit are summed; the summed objective function sensitivity and volume sensitivity are obtained: The symbol <*> α It means summing the integration points α within the macro unit e.

10. The multi-scale structure integrated topology optimization design method according to claim 1, characterized in that: In step S6, the coupling relationship between macroscopic density and microscopic density and the chain derivation rule are used to simplify the iterative update of the optimization solver on the design variables of the macroscopic and microscopic scales into the iterative update of the design variables of the microscopic scale. The error err is calculated at each iteration step update: Where k′ is the current iteration step, c k′-i+1 is the flexibility at the k′-i+1th step, and N is the length of the average interval.

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